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a given ratio to the other; but he has given none concerning magnitudes whereof one together with a given magnitude has a given ratio to the other; though these last occur as frequently in the solution of problems as the first; the reason of which is, that the last may be all demonstrated by help of the first; for, if a magnitude, together with a given magnitude, has a given ratio to another magnitude, the excess of this other above a given magnitude shall have a given ratio to the first, and on the contrary; as we have demonstrated in prop. 14. And for a like reason prop. 15 has been added to the Data. One example will make the thing clear: suppose it were to be demonstrated, that if a magnitude A together with a given magnitude has a given ratio to another magnitude B, that the two magnitudes A and B, together with a given magnitude, have a given ratio to that other magnitude B; which is the same proposition with respect to the last kind of magnitudes ahove mentioned, that the first part of prop. 16, in this edition, is in respect of the first kind: this is shown thus; from the hypothesis, and by the first part of prop. 14, the excess of B above a given magnitude has unto A a given ratio; and, therefore, by the first part of prop. 17, the excess of B above a given magnitude has unto B and A together a given ratio; and by the second part of prop. 14, A and B together with a given magnitude has unto B a given ratio; which is the thing that was to be demonstrated. In like manner the other propositions concerning the last kind of magnitudes may be shown.

PROP. XVI. XVII. In the third part of prop. 10, in the Greek text, which is the 16th in this edition, after the ratio of EC to CB has been shown to be given; from this, by inversion and conversion, the ratio of BC to BE is demonstrated to be given; but without these two steps, the conclusion should have been made only by citing the 6th proposition. And in like manner, in the first part of prop. 11, in the Greek, which in this edition is the 17th, from the ratio of DB to BC being given, the ratio of DC to DB is shown to be given by inversion and composition, instead of citing prop. 7, and the same fault occurs in the second part of the same prop. 11,

PROP. XXI. XXII. These now are added, as being wanting to complete the subject treated of in the four preceding propositions.

PROP. XXIII. This, which is prop. 20, in the Greek text, was separated from prop. 14, 15, 16, in that text, after which it should have been immediately placed, as being of the same kind : it is now put into its proper place; but prop. 21, in the Greek, is left out, as being the same with prop. 14, in that text, which is here prop. 18.

PROP. XXIV. This, which is prop. 13, in the Greek, is now put into its proper

place, having been disjoined from the three following it in this edition, which are of the same kind.

PROP. XXVIII.

This, which in the Greek text is prop. 25, and several of the following propositions, are there deduced from def. 4, which is not sufficient, as has been mentioned in the note on that definition: they are therefore now shown more explicitly.

PROP. XXXIV. XXXVI. Each of these has a determination, which is now added, which occasions a change in their demonstrations.

PROP. XXXVII. XXXIX. XL. XLI. The 35th and 36th propositions in the Greek text are joined into one, which makes the 39th in this edition, because the same enunciation and demonstration serves both: and for the same reason prop. 37, 38, in the Greek, are joined into one, which here is the 40th.

Prop. 37 is added to the Data, as it frequently occurs in the solution of problems; and prop. 41 is added to complete the rest.

PROP. XLII. This is prop. 39, in the Greek text, where the whole construction of prop. 22, of book 1, of the Elements, is put, without need, into the demonstration, but is now only cited.

PROP. XLV.

This is prop. 42, in the Greek, where the three straight lines made use of in the construction are said, but not shown, to be such that any

two of them is greater than the third, which is now done.

PROP. XLVII. This is prop. 44, in the Greek text; but the demonstrations of it is changed into another, wherein the several cases of it are shown, which, though necessary, is not done in the Greek.

PROP. XLVIII. There are two cases in this proposition, arising from the two cases of the third part of prop. 47, on which the 48th depends; and in the composition these two cases are explicitly given.

PROP. LII. The construction and demonstration of this, which is prop, 48, in the Greek, are made something shorter than in that text.

PROP. LIII. Prop. 63, in the Greek text, is omitted, being only a case of prop. 49, in that text, which is prop. 53, in this edition.

PROP. LVIII. This is not in the Greek text, but its demonstration is contained in that of the first part of prop. 54, in that text; which proposition is concerning figures that are given in species': this 58th is true of similar figures, though they be not given in species, and as it frequently occurs, it was necessary to add it.

PROP. LIX. LXI. This is the 54th in the Greek; and the 77th in the Greek, being the very same with it, is left out, and a shorter demonstration is given of prop. 61.

PROP. LXII. This, which is most frequently useful, is not in the Greek, and is necessary to prop. 87, 88, in this edition, as also, though not mentioned, to prop. 86, 87, in the former editions. Prop. 66, in the Greek text, is made a corollary to it.

PROP. LXIV. This contains both prop, 74, and 73, in the Greek text; the first case of the 74th is a repetition of prop. 56, from which it is separated in that text by many propositions; and as there is no order in these propositions, as they stand in the Greek, they are now put into the order which seemed most convenient and natural.

The demonstration of the first part of prop. 73, in the Greek, is grossly vitiated. Dr. Gregory says, that the sentences he has enclosed betwixt two stars are superfluous, and ought to be cancelled ; but he has not observed, that what follows them is absurd, being to prove that the ratio (see his figure) of Ar to T'K is given, which by the hypothesis at the beginning of the proposition is expressly given: so that the whole of this part was to be altered, which is done in this prop. 64.

PROP. LXVII. LXVIII. Prop. 70, in the Greek text, is divided into these two, for the sake of distinctness; and the demonstration of the 67th is rendered shorter than that of the first part of prop. 70, in the Greek, by means of prop. 23, of book 6, of the Elements.

PROP. LXX. This is prop. 62, in the Greek text; prop. 78, in that text, is only a particular case of it, and is therefore omitted.

Dr. Gregory, in the demonstration of prop. 62, cites the 49th prop. dat. to prove that the ratio of the figure AEB to the parallelogram AH is given; whereas this was shown a few lines before : and besides, the 49th prop. is not applicable to these two figures; because AH is not given in species, but is, by the step for which the citation is brought, proved to be given in species.

PROP. LXXIII. Prop. 13, in the Greek text, is neither well enunciated nor demonstrated. The 73d, which in this edition is put in place of

it, is really the same, as will appear by considering (see Dr. Gregory's edition) that A, B, r, E in the Greek text are four proportionals; and that the proposition is to show that a, which has a given ratio to E, is to I, as B is to a straight line to which A has a given ratio; ór, by inversion, that r is to A, as a straight line to which A has a given ratio is to B; that is, if the proportionals be placed in this order, viz. T, E, A, B, that the first r is to A to which the second E has a given ratio, as a straight line to which the third A has a given ratio is to the fourth B; which is the enunciation of this 73d, and was thus changed that it might be made like to that of prop. 72, in this edition, which is the 82d in the Greek text: and the demonstration of prop. 73 is the same with that of prop. 72, only making use of prop. 23, instead of prop. 22, of book 5, of the Elements.

PROP. LXXVII. This is put in place of prop. 79, in the Greek text, which is not a datum, but a theorem premised as a lemma to prop. 80 in that text: and prop. 79 is made cor. 1 to prop. 77, in this edition. Cl. Hardy, in his addition of the Data, takes notice, that in prop. 80, of the Greek text, the parallel KL in the figure of prop. 77, in this edition, must meet the circumference, but does not demonstrate it, which is done here at the end of cor. 3, prop. 77, in the construction for finding a triangle similar to ABC.

PROP. LXXVIII. The demonstration of this, which is prop. 80, in the Greek, is rendered a good deal shorter by help of prop. 77.

PROP. LXXIX. LXXX. LXXXI. These are added to Euclid's Data, as propositions which are often useful in the solution of problems.

PROP. LXXXII. This, which is prop. 60, in the Greek text, is placed before the 83d and 84th, which, in the Greek, are the 58th and 59th, because the demonstration of these two in this edition are deduced from that of prop. 82, from which they naturally follow.

PROP. LXXXVIII. XC. Dr. Gregory, in his preface to Euclid's Works, which he published at Oxford in 1703, after having told that he had supplied the defects of the Greek text of the Data in innumerable places from several manuscripts, and corrected Cl. Hardy's translation by Mr. Bernard's; adds, that the 86th theorem, " or proposition," seemed to be remarkably vitiated, but which could not be restored by help of the manuscripts; then he gives three different translations of it in Latin, according to which, he thinks it may be read; the two first have no distinct meaning, and the third which he says is the best, though it contains a true proposition, which is the 9th in this edition, has no connexion in the least with the Greek text. And it is strange that Dr. Gregory did not observe, that, if prop. 86 was changed into this, the demonstration of the 86th must be cancelled, and another put in

its place: but the truth is, both the enunciation and the demonstration of prop. 86 are quite entire and right, only prop. 87, which is more simple, ought to have been placed before it; and the deficiency which the doctor justly observes to be in this part of Euclid's Data, and which, no doubt, is owing to the carelessness and ignorance of the Greek editors, should have been supplied, not by changing prop. 86, which is both entire and necessary, but by adding the two propositions, which are the 88th and 90th in this edition.

PROP. XCVIII. C. These were communicated to me by two excellent geometers, the first of them by the Right Honourable the Earl of Stanhope, and the other by Dr. Matthew Stewart; to which I have added the demonstrations.

Though the order of the propositions has been in many places changed from that in former editions, yet this will be of little disadvantage, as the ancient geometers never cite the Data, and the moderns very rarely.

AS that part of the composition of a problem which is its construction may not be so readily deduced from the analysis by beginners : for their sake the following example is given, in which the deviation of the several parts of the construction from the analysis is particularly shown, that they may be assisted to do the like in other problems.

PROBLEM.

Having given the magnitude of a parallelogram, the angle of which ABC is given, and also the excess of the square of its side BC above the square of the side AB; to find its sides, and describe it. ·. The analysis of this is the same with the demonstration of the 87th prop. of the Data, and the construction that is given of the problem at the end of that proposition is thus derived from the analysis.

Let EFG be equal to the given angle ABC, and because in the analysis it is said that the ratio of the rectangle AB, BC to the parallelogram AC is given by the 62d prop. dat., therefore, from a point in FE, the perpendicular EG is drawn to FG, as the ratio of FE to EG is the ratio of the rectangle AB, BC to the parallelo

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